Subgame Perfection

In class today we spoke returned to the notion of extensive form games.

You can see a recording of this here.

In class I asked you to write down strategies for a play of the centipede game.

This required writing down an action to take at every possible node of the centipede game. This is what is dictated by the definition of a strategy in an extensive form game

We identified 4 strategies:

• Take at the first opportunity and either take or pass at the second.
• Pass at the second opportunity and either take or pass at the second.

If we consider just these 3 actions then we can reformulate the game as a two player game with the action sets: \(A_1=A_2={\text{TT}, \text{PP}, \text{PT}, \text{TP}}\).

Using this we can now rewrite the utility functions for both players by reading from the tree definition of the centipede game:

\[A = \begin{pmatrix} 2 & 2 & 2 & 2\\ 1 & 4 & 3 & 1\\ 1 & 4 & 4 & 1\\ 2 & 2 & 2 & 2\\ \end{pmatrix} B = \begin{pmatrix} 0 & 0 & 0 & 0\\ 3 & 4 & 5 & 3\\ 3 & 2 & 2 & 3\\ 0 & 0 & 0 & 0\\ \end{pmatrix}\]

We can look at the best response in actions to identify 4 Nash Equilibrium:

\[A = \begin{pmatrix} \underline{2} & 2 & 2 & \underline{2}\\ 1 & \underline{4} & 3 & 1\\ 1 & \underline{4} & \underline{4} & 1\\ \underline{2} & 2 & 2 & \underline{2}\\ \end{pmatrix} B = \begin{pmatrix} \underline{0} & \underline{0} & \underline{0} & \underline{0}\\ 3 & 4 & \underline{5} & 3\\ \underline{3} & 2 & 2 & \underline{3}\\ \underline{0} & \underline{0} & \underline{0} & \underline{0}\\ \end{pmatrix}\]

We see 4 pairs of best responses that correspond to:

\[\{\text{TT}, \text{TT}\}\qquad \{\text{TT}, \text{TP}\} \qquad \{\text{TP}, \text{TT}\}\qquad \{\text{TP}, \text{TP}\}\]

This all give the same outcome but only 1 of them is a Nash equilibrium no matter where we start in the game which is the one we obtained using backwards induction. This is by definition a subgame perfect Nash equilibrium which is the topic of the corresponding chapter.